
Nearest neighbor rounding: Given a set of numbers \(\mathcal {Q}\) , \(\mathsf {Q}(\mathbf {x})\)  returns the nearest number in \(\mathcal {Q}\)  for each coordinate of \(\mathbf {x}\) , i.e., \([\mathsf {Q}(\mathbf {x})]_i=\operatornamewithlimits{\text{argmin}}_{x\in \mathcal {Q}}{|[\mathbf {x}]_i-x|}\) . The uniform quantizers [1]} and logarithmic quantizers [2]} fall in this category by letting \(\mathcal {Q}\)  be the set of numbers spaced evenly on a linear scale and a log scale, respectively. Typically, this class of compressors are biased, and may satisfy only assum3. Particularly, the 1-bit binary quantizer, i.e., \(\mathsf {Q}(x)=0.5\)  for \(x\ge 0\)  and \(\mathsf {Q}(x)=-0.5\)  for \(x< 0\) , satisfies assum3 with \(\delta =0.5\)  and \(p=\infty \) .

Unbiased stochastic quantization: Let \(\text{sgn}(\cdot )\)  and \(|\cdot |\)  be the element-wise sign function and absolute function, respectively. The compressor is given by
\(\mathsf {Q}_u(\mathbf {x})=\frac{\Vert \mathbf {x}\Vert _p\cdot \text{sgn}(\mathbf {x})}{u}\circ \left\lfloor \frac{u|\mathbf {x}|}{\Vert \mathbf {x}\Vert _p}+\xi \right\rfloor ,\) 
where \(\circ \)  denotes the Hadamard product, \(u\ge 1\) , \(\xi \)  is a random vector uniformly sampled from \([0,1]^d\) .
This compressor is unbiased and satisfies assum1 with \(\delta ={d}/{4u^2}\)  for \(p=2\)  [3]}, [4]}, and meets assum3 with \(\delta =\frac{1}{u}\@root p \of {d}\) . A common choice is \(u=2^{l-1}\)  for some integer \(l\ge 1\) , where each coordinate can be encoded with \(l+1\)  bits. Transmitting \(\mathsf {Q}_u(\mathbf {x})\)  needs \((l+1)d+b\)  bits if a scalar can be transmitted with \(b\)  bits with sufficient precision.

Biased quantization: A proper scaling can reduce \(\delta \)  but introduce bias. In particular, \(\mathsf {Q}_b(\mathbf {x})\triangleq \mathsf {Q}_u(\mathbf {x})/\phi \)  with \(\phi > 1\)  is biased. It satisfies assum1 with \(\delta =1-\phi ^{-1}\)  for \(\phi = 1+{d}/{(4u^2)}\)  and \(p=2\)  [5]}, and satisfies assum3 with \(\delta =\frac{1}{u+\@root p \of {d}}\@root p \of {d}\)  for \(\phi = 1+{1}/{u}\cdot \@root p \of {d}\)  and \(\xi ={\mathbf {1}}\) .

Sparsification: Randomly selecting \(l\le d\)  coordinates of \(\mathbf {x}\)  or selecting the largest \(l\)  coordinates in magnitude gives a biased compressor, which meets assum1 with \(\delta =1-{l}/{d}\)  [6]}, and assum3 with \(\delta =(1-{l}/{d})^{1/p}\) .

Compression for multimedia data: The structure of \(\mathbf {x}\)  can be exploited for efficient compression. For example, if \(\mathbf {x}\)  can be viewed as an image, then the standard JPEG method is a good compressor which is biased with an adjustable compression error \(\delta \)  [7]}.


